Applications of stochastic analysis to statistical inference for stationary and non-stationary Gaussian processes

随机分析在平稳和非平稳高斯过程统计推断中的应用

• 批准号: 2311306
• 负责人: Frederi Viens
• 金额: $ 25万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2311306&HistoricalAwards=false
• 关键词: Applications; stochastic; analysis; statistical; inference

Scientists and engineers want to extract information and develop an understanding of the natural world and human society by analyzing data to inform questions and decisions. The data often presents itself as the so-called time series, also known as stochastic processes, evolutions of measurable quantities over periods of time. The variations of these time series can be rather predictable from one period to the next, but less so over longer time intervals covering many periods. There are subtle differences in the nature of various types of these stochastic processes. For instance, the value of a financial stock or index, or the yearly global mean temperature, are buildups, accumulating stochastically over time. But daily returns on stocks or commodities futures, or the category (intensity) of successive Atlantic hurricanes, are of a different nature, typically showing a great deal of independence from one day or one event to the next, featuring a property of stationarity over time after adjusting for trends and seasonality. A critically important question is how some of these time series relate to each other. For instance, are global mean temperatures closely tied to Atlantic hurricane activity? Climate scientists would talk about significant attribution of the latter to the former if the relation is statistically significant. We have discovered that ordinary statistical tools work well to measure attribution when time series are largely stationary, but that the same tools can incorrectly point to a strong attribution when none actually exists, for time series, which are more accumulative. This incorrect attribution phenomenon, measured using a so-called correlation coefficient, occurs more frequently in scientific papers than one would hope. It is known as Yule's "nonsense correlation" in honor of the famed British statistician who first described the possibility empirically in 1926. Our work is the first to quantify exactly how this correlation can behave as a mathematical object, for accumulative time series, and for stationary time series. As a consequence of this award's work, we will provide scientists with demonstrably correct tools for correlations of time series, which will help them measure with great precision whether natural and societal phenomena, such as those described above, are statistically related, or whether they are more likely to be independent of each other. The project will also provide research training opportunities for graduate students. As is a well-accepted direction when developing tools for statistical inference, this award's work will study the properties of statistical tests which detect whether data streams are likely not to be independent. The objects of study are pairs of paths of times series or stochastic processes, and the empirical Pearson-type correlation statistic for any such pair. In particular, the work will apply to observational studies, rather than repeated experiments, since single time series are often the only type of data for any given environmental or economic variable. For stationary stochastic processes, we will derive precise estimates of the empirical correlation's fluctuations, by using calculations involving both exact distribution theory and normal approximations via stochastic analysis. These results will lead directly to proposing principled statistical methods for distinguishing between dependent and independent of pairs of stochastic processes. Next, we will investigate the realm of highly non-stationary paths, including random walks and Brownian motion, how the asymptotics for the empirical correlations deviate strongly from normality, and how to convert this information to the aforementioned application to distinguish between dependence and independence. Much of our work will draw on the distributional properties of classical variance and covariance objects for Gaussian vectors, as a technical aspect of stochastic analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

科学家和工程师希望通过分析数据为问题和决策提出信息来提取信息并发展对自然界和人类社会的理解。数据通常将自己显示为所谓的时间序列，也称为随机过程，在时间内的可测量数量的演变。这些时间序列的变化可以从一个时期到下一个时期是可以预见的，但是在涵盖许多时期的较长时间间隔内较少。这些随机过程的各种类型的性质存在细微的差异。例如，金融股票或指数的价值或年度平均温度是积累，随着时间的流逝而随机积累。但是，股票或商品期货的每日回报，或连续的大西洋飓风的类别（强度）具有不同的性质，通常显示出从一天或一个事件到另一个事件的独立性，在调整了趋势和季节性后，随着时间的推移，随着时间的推移，随着时间的推移是平稳性的。一个至关重要的问题是，其中一些时间序列如何相互关系。例如，全球平均温度是否与大西洋飓风活动紧密相关？如果关系具有统计学意义，气候科学家将谈论后者的重大归因。我们发现，当时间序列基本固定时，普通的统计工具可以很好地衡量归因，但是对于时间序列而言，相同的工具可能会错误地指向强有力的归因，而时间序列更加积累。使用所谓的相关系数测量的这种不正确的归因现象在科学论文中比人们希望更频繁地发生。为了纪念著名的英国统计学家的纪念，它被称为尤尔的“胡说八道相关”，后者在1926年首次从经验上描述了这一可能性。我们的工作是第一个准确量化这种相关性如何作为数学对象，用于累积时间序列的数学对象的工作，以及平稳的时间序列。由于该奖项的作品，我们将为科学家提供明显的时间序列相关工具，这将有助于他们精确地衡量自然和社会现象（例如上述）在统计上是相关的，还是更有可能彼此独立。该项目还将为研究生提供研究培训机会。 正如开发用于统计推断工具时的一个被接受的方向一样，该奖项的工作将研究统计检验的属性，该统计测试检测数据流是否可能不是独立的。研究的对象是时代序列或随机过程的成对，以及任何此类对的经验Pearson型相关统计量。特别是，这项工作将适用于观察性研究，而不是重复实验，因为单个时间序列通常是任何给定环境或经济变量的唯一数据类型。对于固定的随机过程，我们将通过使用随机分析涉及精确分布理论和正常近似的计算来得出经验相关性波动的精确估计。这些结果将直接导致提出原则性的统计方法，以区分依赖性和独立于随机过程对。接下来，我们将研究高度非平稳路径的领域，包括随机步行和布朗运动，经验相关性的渐近性如何与正态性差异很大，以及如何将这些信息转换为上述应用以区分依赖性和独立性。我们的大部分工作都将借鉴高斯向量的经典差异和协方差对象的分布性能，作为随机分析的技术方面。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估的评估来通过评估来支持的。